NUTS_init_tune_mcmc returns the chain and the tuned sampler (so you could continue sampling). Very few people use that so I may remove it as the default, or introduce an alternative API. Currently you can just keep the first value, eg (continuing the above example)
using Random, Distributed
Nchains = 4
rngs = [Random.seed!(i) for i in 1:Nchains];
output = pmap(x -> first(NUTS_init_tune_mcmc(x, ∇P, 1000)), rngs)
works fine (diagnostic output is garbled though, as it is not ready for parallel chains yet).
Conversion to MCMCChain.jl is not supported by my packages (and I don’t know about others). I kind of prefer the bare-bones, modular approach. In practice, IMO the two most useful diagnostics are \hat{R} and ESS, both of which are supported by MCMCDiagnostics.
If you get errors, please make a self-contained MWE and I will look at it.